m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?

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Solution

We observe the following properties of relation R.

$Let R = {(m, n) : m, n \in Z : m - n i s d i v i s i b l e b y 13} Relexivity : Let m be an arbitrary element of Z . Then, m \in R \Rightarrow m - m = 0 = 0 \times 13 \Rightarrow m - m is divisible by 13 \Rightarrow (m, m) is reflexive on Z . Symmetry : Let (m, n) \in R . Then, m - n is divisible by 13 \Rightarrow m - n = 13 p Here, p \in Z \Rightarrow n - m = 13 (- p) Here, - p \in Z \Rightarrow n - m is divisible by 13 \Rightarrow (n, m) \in R for all m, n \in Z So, R is symmetric on Z . Transitivity : Let (m, n) and (n, o) \in R \Rightarrow m - n and n - o are divisible by 13 \Rightarrow m - n = 13 p and n - o = 13 q for some p, q \in Z Adding the above two, we get m - n + n - o = 13 p + 13 q \Rightarrow m - o = 13 (p + q) Here, p + q \in Z \Rightarrow m - o is divisible by 13 \Rightarrow (m, o) \in R for all m, o \in Z So, R is transitive on Z .$

Hence, R is an equivalence relation on Z.

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